Discrete Mathematics with Applications 4th Edition

Published by Cengage Learning
ISBN 10: 0-49539-132-8
ISBN 13: 978-0-49539-132-6

Chapter 6 - Set Theory - Exercise Set 6.3 - Page 373: 27

Answer

$a-\,\,by\,\,\,(commutative\,\,law\,\,for\,\,\cap )\\ b-\,\,by\,\,\,(distributive\,\,law)\\ c-\,\,by\,\,\,(commutative\,\,law\,\,for\,\,\cap) .\\$

Work Step by Step

$For\,\,all\,\,sets\,\,A,\,B,\,and\,C,\\ (A \cup B) \cap C = (A \cap C) \cup (B \cap C).\\ Proof:\,Suppose\,A,\,B,\,and\,C\,are\,any\,sets.\\ Then (A \cup B) \cap C = C \cap (A\cup B)\,\,\,\,\\by (commutative\,\,law\,\,for\,\,\cap )\\ = (C \cap A) \cup (C \cap B)\,\,\,\,\\by (distributive\,\,law)\\ = (A \cap C) \cup (B \cap C)\,\,\,\,\\by (commutative\,\,law\,\,for\,\,\cap) .\\$
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